(PA) is the restrictionPlanetmathPlanetmathPlanetmath of Peano’s axioms to a first order theory of . The only change is that the induction axiomMathworldPlanetmath is replaced by inductionMathworldPlanetmath restricted to arithmetic formulas:

ϕ(0)x(ϕ(x)ϕ(x))xϕ(x))where ϕ is arithmetical

Note that this replaces the single, second-order, axiom of induction with a countably infiniteMathworldPlanetmath schema of axioms.

Appropriate axioms defining +, , and < are included. A full list of the axioms of PA looks like this (although the exact list of axioms varies somewhat from source to source):

  • x(x0) (0 is the first number)

  • x,y(x=yx=y) (the successor function is one-to-one)

  • x(x+0=x) (0 is the additive identity)

  • x,y(x+y=(x+y)) (additionPlanetmathPlanetmath is the repeated application of the successor function)

  • x(x0=0)

  • x,y(x(y)=xy+x) (multiplication is repeated addition)

  • x(¬(x<0)) (0 is the smallest number)

  • x,y(x<yx<yx=y)

  • ϕ(0)x(ϕ(x)ϕ(x))xϕ(x))where ϕ is arithmetical

Title PA
Canonical name PA
Date of creation 2013-03-22 12:56:36
Last modified on 2013-03-22 12:56:36
Owner Henry (455)
Last modified by Henry (455)
Numerical id 10
Author Henry (455)
Entry type Definition
Classification msc 03F30
Synonym Peano arithmetic
Synonym first order Peano arithmetic
Related topic PeanoArithmetic